We consider two types of folding applied to equilateral plane graph
linkages. First, under continuous folding motions, we show how to reconfigure
any linear equilateral tree (lying on a line) into a canonical
configuration. By contrast, such reconfiguration is known to be impossible
for linear (nonequilateral) trees and for (nonlinear) equilateral trees.
Second, under instantaneous folding motions, we show that an equilateral plane
graph has a noncrossing linear folded state if and only if it is bipartite.
Not only is the equilateral constraint necessary for this result, but we show
that it is strongly NP-complete to decide whether a (nonequilateral) plane
graph has a linear folded state. Equivalently, we show strong NP-completeness
of deciding whether an abstract metric polyhedral complex with one central
vertex has a noncrossing flat folded state with a specified “outside
region”. By contrast, the analogous problem for a polyhedral manifold
with one central vertex (single-vertex origami) is only weakly
NP-complete.